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How can I add an interval, to arrive at one answer here. My answer should be between 0, pi/2

How can I add an interval, to arrive at one answer here. My answer should be between 0, pi/2

I tried solving

solve(cos(2*t)==sin(t),t)

I got

[cos(2*t) == sin(t)]

But this shouldn't be the value. I know from the graphs that the value is numeric.

I have also noticed that sagemath is not good for solving trig identities. Is this true?

I have a simple equation to solve

solve(ln(x) -2 > 0,x)

The actual value should be $x > e^2$

but sagemath gives

[[log(x) - 2 > 0]]

How do I get actual value?

when I run the expression

solve([x>=3, x >=5, x < 8],x)

It returns

[[5 < x, x < 8], [x == 5]]

But I'm hoping that it would return

[[5 <= x, x < 8]]

How do I get my expected value?

Is there a way to test programmatically if a function is continuous?

e.g $f(x) = \frac{1}{x} $

I'm looking for a function is_continuous(f) = False

I'm trying to differentiate an implicit expression

$x e^{y} = x -y$

This is my sagemath code

x = var('x')
f(x,y)= x*e**y - x + y
show(diff(f))

Sagemath Answer is $\left( x, y \right) \ {\mapsto} \ \left(e^{y} - 1,\,x e^{y} + 1\right)$

But the actual answer is $\frac{1 - e^{y}}{x e^{y} + 1}$

How do I get the actual answer using sagemath?

I'm trying to differentiate an implicit expression

$x e^{y} = x -y$

This is my sagemath code

x = var('x')
f(x,y)= x*e**y - x + y
show(diff(f))

Sagemath Answer is $\left( x, y \right) \ {\mapsto} \ \left(e^{y} - 1,\,x e^{y} + 1\right)$

But the actual answer is $\frac{1 - e^{y}}{x e^{y} + 1}$

How do I get the actual answer using sagemath?